Infinite series in cohomology: Attractors and Conley index
Abstract.
In this paper we study the cohomological Conley index of arbitrary isolated invariant continua for continuous maps by analyzing the topological structure of their unstable manifold. We provide a simple dynamical interpretation for the first cohomological Conley index, describing it completely, and relate it to the cohomological Conley index in higher degrees. A number of consequences are derived, including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3.
Our approach exploits certain attractorrepeller decomposition of the unstable manifold, reducing the study of the cohomological Conley index to the relation between the cohomology of an attractor and its basin of attraction. This is a classical problem that, in the present case, is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated. To address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology: if is a metric space and is a compact, global attractor for a continuous map , we show how to interpret series of the form as endomorphisms of the cohomology group of the pair .
Key words and phrases:
AlexanderSpanier cohomology, attractors, Conley index, fixed point index, filtration pairs2000 Mathematics Subject Classification:
37C25, 37B30, 54H251. Introduction
The Conley index is a powerful topological invariant of dynamical systems that has proved to be very useful. It can be thought of as a farreaching generalization of the fixed point index to (almost) arbitrary compact invariant sets , and in fact both indices satisfy formally similar properties. The only requirement on is that it be isolated or locally maximal, which means that it has a compact neighbourhood such that is the maximal invariant subset of (in that case is called an isolating neighborhood of ). The Conley index was first introduced by Conley [conley1] in the context of continuous dynamical systems under the name “Morse index” and extended later on by several authors (see for instance [franksricheson], [handbookconley], [Mro1], [robinsalamon1] or [Szy2]), to discrete dynamical systems. In either case the index is homotopical in nature, but it has homological and cohomological versions that are often more convenient to work with. For a complete introduction to the theory of Conley index we refer the reader to [handbookconley].
In this paper we shall be concerned with the (co)homological Conley index for a discrete dynamical system generated by the iteration of some continuous map . The precise definition is somewhat involved and will be recalled later on, but for the purposes of this Introduction the following rough idea will suffice: for each dimension the th homological Conley index of an isolated invariant set is an endomorphism of a finite dimensional vector space over a field ; intuitively, captures the action of on a neighbourhood of at the level of –dimensional homology. The homology is computed with coefficients in , which we will take to be or . By considering the eigenvalues or the trace of one obtains even simpler, yet still powerful, numerical invariants. Similar considerations apply to the th cohomological Conley index .
The following two results from [HCR] are our starting point (recall that a compact set is called acyclic if its Čech homology groups of degree vanish):
Theorem A.
Let be a homeomorphism defined on an open subset of and let be an isolated invariant acyclic continuum. Then there exist a finite set and a map such that the trace of the first homological Conley index satisfies
In particular, .
Theorem B.
Let be a homeomorphism defined on an open subset of and let be an isolated invariant acyclic continuum. If then for every .
The rather difficult proofs of these theorems involve a combinatorial description of the first homological Conley index that is ultimately related to how acts on the components of the exit set of an isolating neighbourhood of (the exit set consists of the points in whose image under lies outside ). Several questions arise naturally:

Is it possible to provide a dynamical interpretation of the map and the set ?

Is it possible to generalize Theorems A and B to non acyclic continua ? This suggests that we turn to the cohomological Conley index and use Čech cohomology , which is the preferred theory for sets with a complicated local structure, and indeed henceforth we shall do so. (By the universal coefficient theorem, acyclic sets also have vanishing Čech cohomology in every degree ).

Is it possible to extend the results for arbitrary continuous maps , rather than homeomorphisms, and possibly more general phase spaces?
In this paper was to provide answers to all these questions, obtaining suitable generalizations of Theorems A and B and deriving from them a number of corollaries concerning the fixed point index of isolated invariant continua (these are discussed in Section LABEL:sec:corollaries). The original proofs in [HCR] involved a very delicate “woodworking” with homology classes in an isolating neighbourhood of , but here we shall adopt a radically different approach which is much more algebraic in nature. It accommodates naturally the non acyclic case and involves only dynamically meaningful objects that are canonically associated to (namely, its unstable manifold), providing a straightforward interpretation of and . In order to be able to carry out this approach we develop a notion of summation of infinite series in cohomology that may be of independent interest.
Main results
In order to state our main results we need to recall a construction due to Robbin and Salamon [robinsalamon1]. We will be rather informal, postponing precise definitions to Section LABEL:sec:background. Given an invariant set for a homeomorphism, its unstable manifold is defined as the subset of phase space consisting of those points whose negative semitrajectory approaches , in the sense that it eventually enters any neighbourhood of . The unstable manifold (which, in spite of its name, need not be a manifold) is an invariant set that contains . Robbin and Salamon introduced the socalled intrinsic topology on , which is generally finer than the one that it inherits from the phase space and, roughly speaking, makes two points close if and only if their backward orbits remain close to each other for a large number of iterates. The unstable manifold endowed with the intrinsic topology has the following properties:

is a locally compact, Hausdorff space. In particular it has an Alexandroff compactification , whose point at infinity we shall denote by .

generates a discrete dynamical system on which can be trivially extended to all of declaring to be a fixed point.

The pair is an attractorrepeller Morse decomposition of .
We will show in Section LABEL:sec:background that, under very general circumstances, the induced homomorphism is equivalent to the cohomological Conley index of . Notice that this expresses the cohomological Conley index in terms of a canonical object associated to (its unstable manifold), as mentioned earlier. This justifies our interest in the cohomology of and the action of on it.
Now consider the set . In general this set may have infinitely many connected components, or rather quasicomponents, as it will turn that it is advantageous to work with the latter. This owes to the fact that the unstable manifold may have very complicated topological structure, as illustrated by the construction given in [Art]. These quasicomponents are typically called branches of the unstable manifold. We say that a quasicomponent is essential if its closure in has a nonempty intersection with and ; intuitively, it “joins” and . Clearly, permutes the essential quasicomponents of among themselves.
Since we want to consider the general case when is not necessarily a homeomorphism but merely a continuous map, the preceding discussion will need some revision. The construction of Robbin and Salamon carries over with appropriate modifications and leads naturally to the replacement of and with a homeomorphism on and a compact invariant set in such a way that properties (i) to (iii) above remain true. These new objects and are closely related but not exactly equal to and but, since the differences are largely inconsequential, we suggest that the reader assumes to be a homeomorphism and simply ignores the tildes for the time being.
Now we can state two of the two main results in this paper, which describe the cohomology of in terms of the essential quasicomponents. In degree one it is actually convenient to use as a basepoint and work with the relative cohomology group because it has a more symmetric description than . We have the following:
Theorem 1.
Let be a continuous map and assume has finitely many connected components. Then there are only finitely many essential quasicomponents in and the cohomology group can be identified with the –vector space having the set as a basis. Moreover, under this identification the map simply becomes the permutation .
This result suggests that we introduce the following notation: will denote the set of essential quasicomponents of and will denote the permutation induced by on .
For higher degrees, our second main theorem roughly states that the cohomology of is entirely determined by that of the essential quasicomponents alone:
Theorem 2.
Let be a continuous map and assume has finitely many connected components. In any degree such that is finite dimensional, there is an isomorphism that commutes with . Here denotes the closure of in the quotient space that results from by collapsing to a single point.
The description given by the theorems above holds for the iterates () of as well. Upon replacing with the set is still isolated for this new dynamical system, the compactified unstable manifold and its subset remain essentially the same, and the homeomorphism induced on can be identified with . (This is almost immediate when is a homeomorphism but needs a more careful proof for continuous maps. Statements can be found in Subsection LABEL:subsec:convention). As a consequence, the set of essential quasicomponents for can be identified with and the permutation induced by on is then . With these changes, Theorems 1 and 2 extend to .
It is clear from Theorem 1 that the trace of on is the number of essential quasicomponents that are fixed by or, equivalently, the number of fixed points of . Since the trace of the –cohomological Conley index of coincides with that of , it is just a matter of using the long exact sequence of the triple and the additivity of the trace to obtain the following generalization of Theorem A:
Corollary 3.
Let be a continuous map and let be an arbitrary isolated invariant continuum. Denote by the map induced by the inclusion in onedimensional cohomology. Then the trace of the first cohomological Conley index of is given by
unless is an asymptotically stable attractor, in which case only the third summand should be retained.
If (in particular, if is acyclic) then the third term vanishes, yielding an expression which is formally identical to that of Theorem A. In our context, however, we have a clear dynamical interpretation for both and .
The eigenvalues of can also be easily described in similar terms. Since the set of essential quasicomponents is finite, the action of partitions it into disjoint, irreducible cycles of the form It is straightforward to check from Theorem 1 that each of these contributes a factor to the characteristic polynomial of on . Thus, when coefficients are taken in each cycle of quasicomponents of minimal period contributes eigenvalues of that constitute precisely a complete set of th roots of unity. Again through the exact sequence for the pair it follows that every eigenvalue of is either one of or one of the roots of unity just described.
The generalization of Theorem B reads as follows:
Corollary 4.
Let be a continuous map and let be an arbitrary isolated invariant continuum. Denote by and the homomorphisms induced by the inclusion in – and –dimensional cohomology. Assume that ; that is, sends every essential quasicomponent onto a different one. Then
in any degree such that and are finitely generated.
The proof of Corollary 4 is a simple consequence of Theorem 2 but involves a minor technical point concerning the relation between and . We shall discuss this in detail in Section LABEL:sec:background and prove Corollaries 3 and 4 in Section LABEL:sec:proofs34.
Corollary 4 may not be very appealing aesthetically, but it has a neat consequence concerning the fixed point index of on . This very classical invariant is an integer that provides an algebraic measure of the amount of fixed points that a continuous map has. In particular, a non–zero index implies the existence of fixed points. The celebrated Lefschetz–Hopf theorem states that for compact triangulable spaces the index is equal to the socalled Lefschetz number of , which is defined as the alternated sum of the traces of the maps induced by on the singular (co)homology groups and usually denoted by . This result was later on extended by Lefschetz himself to the case where is a compact absolute neighborhood retract (see [Brown] or [JM]). It is known, however, that it does not generally hold when has bad local topological features. The interest of the following corollary is that it provides another instance where the Lefschetz–Hopf theorem holds even though no direct assumption about the topology of is made; in fact, the latter may well be very complicated:
Corollary 5.
Let be a continuous map defined in a manifold and let be an isolated invariant continuum having finitely generated Čech cohomology. Then, the (Čech) Lefschetz numbers , where denotes the map induced by in the th Čech cohomology groups of , are well defined. Assume that , so that sends every essential quasicomponent onto a different one, then .
More generally, if we denote by the greatest common divisor of the periods of the action of in the set of essential quasicomponents, then if is not multiple of any of those periods and otherwise.
Notice that the condition that is automatically satisfied when is an asymptotically stable attractor, since in that case the compactified unstable manifold reduces to the disjoint union of and , there are no essential quasicomponents and the hypotheses are trivially satisfied. Thus for attractors the proposed generalization of Lefschetz–Hopf theorem holds true.
As mentioned earlier, the action of partitions into disjoint irreducible cycles of the form Written out in full, this means that , so in particular all the quasicomponents involved in a cycle are homeomorphic to each other. This fact can be exploited in several ways to extract information about certain numerical invariants. For instance, , for , and this implies that equations in Corollaries 3 and 4 are valid no matter how acts on . This consequence can be easily tracked in the proofs of those corollaries in Section LABEL:sec:proofs34. The second statement of Corollary 5 provides an example of this technique (another example is furnished by Theorem 6 below):
Proof of Corollary 5.
This result is a direct application of the Lefschetz–like formula (see [Franks2], [HCR], [LCRPS] or [mrozekLefschetz])
Upon replacing the formulae for given in Corollaries 3 and 4 and using the remark above a telescopic cancellation ( if ) occurs that eliminates all the traces of , yielding the equality (valid if ). The case in which is an attractor requires special treatment because the term does not vanish, it is equal to 1 (cf. Section LABEL:sec:corollaries), but this contribution cancels with the particular formula in Corollary 3. ∎
Lefschetz–Hopf theorem sharpens the celebrated original result of Lefschetz, usually referred to as Lefschetz theorem, which states that if then . This theorem has been proven in several instances but does not hold for general arbitrary continua (non–ANR). In 1935 Borsuk [Bor01] (see also [BarSad], [Bing], [Hag] and references therein) constructed a locally connected acyclic continuum of without the fixed point property, that is, such that there is a continuous map without fixed points. From the previous discussion we learn, for instance, that it is impossible to embed (or any other counterexample to the Lefschetz theorem) in in such a way that can be extended to a continuous map defined on a neighborhood of and such that is an attractor for . Indeed, suppose such an embedding would be possible. Then and would satisfy the hypothesis of Corollary 5 and the condition would be automatically satisfied so , which would imply that and thus has fixed points in .
Theorem 6.
Let be an isolated invariant continuum for a homeomorphism . Denote by the greatest common divisor of the periods of the essential quasicomponents of . Assume has finitely generated cohomology in all higher degrees so that is well defined. Then
In specific examples the Conley index is usually explicitly computable whereas the isolated invariant set itself might be more elusive. We shall show in Section LABEL:sec:background that can in turn be computed from the Conley index of , and so the theorem above shows that it is possible to extract some information about the Euler characteristic of the “unobservable” from its “observable” Conley index.
Proof of Theorem 6.
(We do not need to distinguish between and or and because is assumed to be a homeomorphism). Clearly the number of essential quasicomponents in must be divisible by , and then Theorem 1 implies that . Here denotes the th Betti number; that is, the dimension of the corresponding Čech cohomology group (vector space) . As for the higher Betti numbers, for we have where the last equality follows from Theorem 2. Since all the essential quasicomponents in a cycle are homeomorphic to each other and so are their closures in , the all have the same Betti numbers and contribute equally to the above sum. The number of summands corresponding to each cycle is divisible by , so it follows that too. Finally, because, by Proposition LABEL:prop:reach, each quasicomponent of reaches or (or both, if essential). Thus . The theorem follows from this and the additivity of the Euler characteristic, which implies that . ∎
Most of the results in this paper are stated for connected for simplicity. However, since all of them follow from Theorems 1 and 2, which are valid when has finitely many connected components, they all have appropriate generalizations to this slightly more general case. However, a qualitative leap takes place when one considers sets having infinitely many connected components: as the reader will see there arise difficulties which, although algebraic in nature, seem to reflect new dynamical phenomena that cannot occur when is connected or has only finitely many connected components. For instance, we will prove the following result:
Theorem 7.
Assume that the first cohomological Conley index has a nonzero eigenvalue that is not a root of unity. Then at least one of the following holds:

has infinitely many connected components.

has as an eigenvalue.
In particular, if the phase space is and is a homeomorphism, then must have infinitely many connected components because (ii) cannot hold.
Unfortunately our present understanding of the interplay between algebra and dynamics when has infinitely many connected components is still rather poor, so we will have to content ourselves with a characterization of the eigenvalues and eigenvectors of on (not nearly as neat as that of Theorem 1) and a proof of Theorem 7.
Our approach to the problem
Having given a taste of the sort of results that will be obtained in this paper, we will now present the general reasoning behind the proof of Theorem 1 (we warn the reader that the ideas presented here will turn out to be too simplistic and are therefore not in their final form). For this informal discussion we shall dispense with the notational distinction between , , and .
Begin by writing as the union of the two open invariant sets
and consider the associated Mayer–Vietoris sequence
(1) 